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Flow-induced vibrations of a rotating cylinderRémi Bourguet, David Lo Jacono
To cite this version:Rémi Bourguet, David Lo Jacono. Flow-induced vibrations of a rotating cylinder. Journal of FluidMechanics, Cambridge University Press (CUP), 2014, vol. 740, pp. 342-380. <10.1017/jfm.2013.665>.<hal-00947888>
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To link to this article : DOI:10.1017/jfm.2013.665 URL : http://dx.doi.org/10.1017/jfm.2013.665
To cite this version : Bourguet, Rémi and Lo Jacono, David Flow-induced vibrations of a rotating cylinder. (2014) Journal of Fluid Mechanics, vol. 740 . pp. 342-380. ISSN 0022-1120
Flow-induced vibrations of a rotating cylinder
Rémi Bourguet† and David Lo JaconoInstitut de Mécanique des Fluides de Toulouse, CNRS, UPS and Université de Toulouse,
31400 Toulouse, France
The flow-induced vibrations of a circular cylinder, free to oscillate in the cross-flowdirection and subjected to a forced rotation about its axis, are analysed by means oftwo- and three-dimensional numerical simulations. The impact of the symmetrybreaking caused by the forced rotation on the vortex-induced vibration (VIV)mechanisms is investigated for a Reynolds number equal to 100, based on thecylinder diameter and inflow velocity. The cylinder is found to oscillate freely up toa rotation rate (ratio between the cylinder surface and inflow velocities) close to 4.Under forced rotation, the vibration amplitude exhibits a bell-shaped evolution asa function of the reduced velocity (inverse of the oscillator natural frequency) andreaches 1.9 diameters, i.e. three times the maximum amplitude in the non-rotating case.The free vibrations of the rotating cylinder occur under a condition of wake–bodysynchronization similar to the lock-in condition driving non-rotating cylinder VIV.The largest vibration amplitudes are associated with a novel asymmetric wake patterncomposed of a triplet of vortices and a single vortex shed per cycle, the T+S pattern.In the low-frequency vibration regime, the flow exhibits another new topology, theU pattern, characterized by a transverse undulation of the spanwise vorticity layerswithout vortex detachment; consequently, free oscillations of the rotating cylinder mayalso develop in the absence of vortex shedding. The symmetry breaking due to therotation is shown to directly impact the selection of the higher harmonics appearingin the fluid force spectra. The rotation also influences the mechanism of phasingbetween the force and the structural response.
Key words: flow–structure interactions, vortex streets, wakes
1. IntroductionFlow-induced vibrations (FIV) of flexible or flexibly mounted bodies with bluff
cross-section are encountered in a great variety of physical systems, from theoscillations of plants in wind to the vibrations of risers and mooring lines immersedin ocean currents. Such vibrations cause increased fatigue damage and sometimesfailure of the structures. Their modelling, prediction and the elaboration of vibrationreduction techniques require a detailed understanding of the underlying fluid–structureinteraction mechanisms. The impact of FIV in several civil, wind, offshore and nuclearengineering applications has motivated a number of studies, as collected in Blevins(1990), Naudascher & Rockwell (1994) and Païdoussis, Price & de Langre (2010).
†Email address for correspondence: [email protected]
4 5 6 7 8 90
0.1
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0.5
0.6
0.05
0.15
0.25
0.10
0.20
0 1.0 2.0 3.0 4.03.52.51.50.5
(a () b)
FIGURE 1. (a) Maximum amplitude of vibration for a flexibly mounted non-rotatingcylinder (α = 0) as a function of the reduced velocity. (b) The r.m.s. value of thecross-flow force coefficient fluctuation for a rigidly mounted rotating cylinder as a functionof the rotation rate.
Vortex formation downstream of a bluff structure induces unsteady forces on thebody, which can result in structural vibrations. Vortex-induced vibrations (VIV) area type of FIV that have been extensively investigated through the canonical problemof a rigid circular cylinder forced or free to oscillate within a cross-flow (Bishop &Hassan 1964; Bearman 1984, 2011; Williamson & Roshko 1988; Mittal & Tezduyar1992; Hover, Techet & Triantafyllou 1998; Carberry, Sheridan & Rockwell 2001; Jeon& Gharib 2001; Sarpkaya 2004; Williamson & Govardhan 2004; Klamo, Leonard &Roshko 2006; Lucor & Triantafyllou 2008; Dahl et al. 2010; Navrose & Mittal 2013).The axial symmetry of a circular cylinder allows VIV to be examined independentlyof other types of FIV, such as galloping, which is associated with aerodynamicallyunstable cross-sections (Païdoussis et al. 2010). The typical cross-flow response of anelastically mounted circular cylinder is illustrated in figure 1(a), where the maximumamplitude of vibration, normalized by the cylinder diameter, is plotted as a functionof the reduced velocity, defined as the inverse of the oscillator natural frequencynon-dimensionalized by the inflow velocity and the cylinder diameter. Self-excitedvibrations appear over a well-defined range of reduced velocities. In this range, thefrequency of vortex formation and the frequency of body oscillation coincide; thiscondition of wake–body synchronization is referred to as lock-in. Under lock-in,the vortex shedding frequency can depart substantially from the Strouhal frequency,i.e. the shedding frequency downstream of a stationary cylinder; also, the vibrationfrequency can shift considerably away from the natural frequency, owing to changes inthe value of the effective added mass (Williamson & Govardhan 2004). The maximumamplitude of VIV responses is generally of the order of one cylinder diameter; it is,however, influenced by structural properties (e.g. mass-damping parameter) and by theReynolds number (Re) (Bearman 2011), and it can reach two diameters at Re≈ 105
(Raghavan & Bernitsas 2011). In the above-mentioned study and in the present work,the Reynolds number is based on the cylinder diameter and oncoming flow velocity.
VIV can still occur when the symmetry of the body is broken, for instance byconsidering a square cross-section (Bearman et al. 1987; Zhao, Cheng & Zhou2013). Under broken symmetry, flexibly mounted bodies may also be subjected tothe phenomenon of galloping, which is an aerodynamic instability characterized bylow-frequency oscillations that increase in amplitude unboundedly with the reduced
velocity (Païdoussis et al. 2010). Galloping is driven by the instantaneous angleof attack between the body and the flow, which results in an asymmetric pressuredistribution. It does not involve a mechanism of lock-in between the oscillationand vortex formation as opposed to VIV and can thus often be predicted throughquasi-steady theory. However, depending on the structure properties, VIV andgalloping regions may sometimes overlap (Bearman et al. 1987; Corless & Parkinson1988); the interaction between these two types of FIV leads to intermediate vibrationregimes with larger-amplitude responses (Nemes et al. 2012).
In the present study, symmetry breaking is introduced through body actuation,by applying a forced rotation to the flexibly mounted circular cylinder. Suchconfigurations may occur in practical applications, as for instance in offshoreengineering where drilling risers without casing may be exposed to ocean currents.The possible enhancement of the cylinder free oscillations by the forced rotationmay also have implications in the domain of flow energy harvesting. Since the earlywork of Prandtl (1926), the case of a rigidly mounted circular cylinder subjected toforced rotation within a cross-flow has been the object of many studies, partly dueto its potential applications to wake control (e.g. Modi 1997). Different flow regimeshave been identified depending on the rotation rate α, defined as the ratio betweenthe cylinder surface velocity and the oncoming flow velocity. Over a wide range ofReynolds numbers, previous experimental and numerical work has shown that thealternate vortex shedding associated with the von Kármán instability vanishes abovea rotation rate α ≈ 2 (Coutanceau & Ménard 1985; Badr et al. 1990; Chew, Cheng& Luo 1995; Kang, Choi & Lee 1999; Stojkovic, Breuer & Durst 2002; Mittal &Kumar 2003; Pralits, Brandt & Giannetti 2010). A secondary region of unsteadywake where one-sided vortices form at much lower frequency has been reported overa small range of rotation rates, for α > 4 (Mittal & Kumar 2003; Stojkovic et al.2003; Pralits et al. 2010). The imposed rotation alters the flow three-dimensionaltransition. Through direct numerical simulation and stability analysis, El Akoury et al.(2008) and Rao et al. (2013b) have shown that the rotation increases the criticalReynolds number of the secondary instability associated with the spanwise undulationof the von Kármán vortices. The three-dimensional transition scenario involves wakemodes similar to those identified in the non-rotating cylinder case (i.e. modes Aand B (Williamson 1988)) but also new steady and unsteady modes, depending onthe rotation rate and Reynolds number (Rao et al. 2013b). Over a range of highrotation rates, Mittal (2004) and Meena et al. (2011) presented evidence that thethree-dimensional transition may occur through flow patterns comparable to thoseobserved due to centrifugal instabilities in flows between two concentric rotatingcylinders. They showed that such three-dimensional patterns appear in both steadyand unsteady flow regimes. On the basis of linear stability analyses, Pralits, Giannetti& Brandt (2013) and Rao et al. (2013a) recently emphasized that the imposed rotationmay cause three-dimensional transition down to Re≈ 30; at Re= 100, they reporteda critical rotation rate close to 3.7.
The passage from unsteady to steady flow at α ≈ 2 is accompanied by thesuppression of the fluid force fluctuations, as illustrated in figure 1(b), where theroot mean square (r.m.s.) value of the cross-flow force coefficient fluctuation exertedon a rigidly mounted rotating cylinder is plotted as a function of the rotation rate.The question arises whether the structural vibrations will be cancelled by the rotationin this range of α once the cylinder is free to oscillate.
The problem of a flexibly mounted circular cylinder with imposed rotation in across-current has received much less attention than the rigidly mounted body case.
Through experiments and numerical simulations, Stansby & Rainey (2001) examinedthe case of a rotating circular cylinder free to vibrate in both the in-line and cross-flowdirections, in the Reynolds number range 200–4700. For α ∈ [2, 5] and dependingon the reduced velocity, they reported large oscillations of the cylinder, with peakamplitudes higher than 10 diameters. These low-frequency galloping-like responsesoccur without lock-in; instead, vortex formation leads to high-frequency oscillationsof the force coefficients that are superimposed on the low-frequency oscillationsassociated with the cylinder responses. Yogeswaran & Mittal (2011) confirmed theseobservations at α = 4.5 and Re = 200, and provided flow visualizations emphasizingthe absence of synchronization between the cylinder large-amplitude low-frequencyoscillations and the vortex formation. Previous work has focused on two-degree-of-freedom oscillators, which have been shown to be prone to galloping responses whenthe cylinder is subjected to forced rotation. Hence, the behaviour of the system isunder question when the rotating cylinder is free to oscillate only in the cross-flowdirection. In addition, previous studies did not analyse the effect of the rotation in theregime where wake and body oscillations are synchronized; the influence of forcedrotation on VIV phenomenon remains to be elucidated. The possible consequences,on the cylinder responses, of the flow three-dimensional transition at high rotationrates also need to be clarified. These aspects are addressed in the present work.
The impact of symmetry breaking, through imposed rotation of the cylinder, on thefluid–structure interaction mechanisms previously examined in the non-rotating case isinvestigated in this study by means of a joint analysis of the body responses, wakepatterns and fluid forces. A flexibly mounted circular cylinder, free to oscillate inthe cross-flow direction, is considered over a wide range of reduced velocities androtation rates. The range of rotation rates includes the transition region where vortexshedding is suppressed by the rotation in the rigidly mounted body case. The Reynoldsnumber is chosen equal to 100, which ensures a two-dimensional flow over most ofthe parameter space, except in the range of high rotation rates. The parametric study isbased on two- and three-dimensional numerical simulations of the flow past a flexiblymounted body.
The paper is organized as follows. The physical fluid–structure model and thenumerical method are presented in § 2. The cylinder responses are quantified in § 3.The flow patterns encountered in the wake of the flexibly mounted rotating cylinderare described in § 4. The fluid forces are analysed in § 5. The three-dimensionaltransition in the flow is examined in § 6. Finally, the main findings of the presentwork are summarized in § 7.
2. Formulation and numerical methodThe physical configuration and its modelling are described in § 2.1. The numerical
method employed and its validation are presented in § 2.2.
2.1. Fluid–structure systemA sketch of the physical configuration is presented in figure 2. The elastically mountedrigid cylinder has a circular cross-section; it is immersed in a cross-flow that is parallelto the x axis and normal to the cylinder axis (z axis). The Reynolds number based onthe oncoming flow velocity (U) and cylinder diameter (D), Re= ρf UD/µ, where ρfand µ denote the fluid density and viscosity, is set equal to 100. The flow is assumedto be incompressible. At Re= 100, the flow remains two-dimensional over most of theparameter space investigated; therefore, the two-dimensional Navier–Stokes equations
k
D
z x
y
FIGURE 2. Sketch of the physical configuration.
are generally employed to predict the flow dynamics. In order to study the flowthree-dimensional transition at high rotation rates, the three-dimensional Navier–Stokesequations are also considered at selected points of the parameter space. In the three-dimensional case, an aspect ratio of L/D= 10, where L is the cylinder length in thespanwise direction (z axis), is selected as a reasonable balance between the wavelengthof the flow three-dimensional pattern (of the order of (1–2)D) and the numerical cost.
The cylinder is fixed in the in-line direction (x axis) and free to oscillate in thecross-flow direction (y axis). The structure/fluid mass ratio, m= ρc/ρf D2, where ρc isthe body mass per unit length, is set to a value of 10. The structural stiffness anddamping ratio are designated by k and ξ . In the following, all the physical variablesare non-dimensionalized by the cylinder diameter and the oncoming flow velocity.The non-dimensional cylinder displacement, velocity and acceleration are denoted byζ , ζ and ζ , respectively. The sectional in-line and cross-flow force coefficients aredefined as Cx = 2Fx/ρf DU2 and Cy = 2Fy/ρf DU2, where Fx and Fy are the in-lineand cross-flow dimensional sectional fluid forces. The body dynamics is governedby a forced second-order oscillator equation, which can be expressed as follows, innon-dimensional formulation:
ζ + 4πξ
U?ζ +
(2π
U?
)2
ζ = Cy
2m. (2.1)
Here U? denotes the reduced velocity, U? = 1/fn, where fn is the non-dimensionalnatural frequency in vacuum, fn =D/2πU
√k/ρc. The structural damping is set equal
to zero (ξ =0) to allow maximum-amplitude oscillations. In the three-dimensional flowcase, the span-averaged fluid force coefficient is considered in (2.1).
The cylinder is subjected to a forced, anticlockwise, steady rotation about its axis.The imposed rotation is controlled by the rotation rate α =ΩD/2U, where Ω is theangular velocity of the cylinder.
The present parametric study focuses on α and U?. Previous work concerning rigidlymounted rotating cylinders has shown that the critical value of the rotation rate for
suppression of the vortex shedding at Re= 100 is α≈ 1.8 (Kang et al. 1999; Stojkovicet al. 2003). A range of rotation rates including this transition is considered, namelyα ∈ 0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 3.75, 4.0. The reduced velocity is varied from4 to 34 with a step equal to 0.5.
2.2. Numerical simulation methodThe coupled fluid–structure system is solved by the parallelized code Nektar, whichis based on the spectral/hp element method (Karniadakis & Sherwin 1999). Theversion of the code employs a Jacobi–Galerkin formulation in the (x, y) plane anda Fourier expansion in the spanwise (z) direction (for three-dimensional simulations).A boundary-fitted coordinate formulation is used to take into account the cylinderdisplacement. Details concerning the numerical method, including the time integrationschemes, and its implementation have been reported in Newman & Karniadakis (1997)and Evangelinos & Karniadakis (1999) for similar configurations.
A non-dimensional time step ranging from 0.001 to 0.005 is selected, depending onthe rotation rate. As noted by Mittal & Kumar (2003), the size of the computationaldomain may substantially impact the simulation results, especially at high rotationrates. To avoid any spurious blockage effects due to domain size, a large rectangulardomain is considered in the present simulations: 350D downstream; and 250D in front,above and below the cylinder. A no-slip condition is applied on the cylinder surface.The free-stream value is assigned for the velocity at the upstream boundary. At thedownstream boundary, a Neumann-type boundary condition is used. Flow periodicityconditions are employed on the upper and lower boundaries, as well as on the side(spanwise) boundaries in the three-dimensional case. A two-dimensional grid of 3975spectral elements is used in the (x, y) plane. The evolutions of the time-averagedforce coefficients as functions of the spectral element polynomial order, for a rigidlymounted cylinder at α = 4 (the largest rotation rate considered in this paper) and fora flexibly mounted cylinder at (α, U?) = (3.75, 13) (i.e. in the region of maximumvibration amplitude, as shown in § 3), are plotted in figure 3. In the former case, therotation cancels the vortex shedding and the flow is steady, thus Cx=Cx and Cy=Cy,where the over-bar denotes time-averaged values. It can be observed that an increasefrom order four to five has no significant influence on the results; hence, a polynomialorder equal to four is selected. For the three-dimensional simulations, 64 complexFourier modes (i.e. 128 planes) are employed in the spanwise direction.
The simulation approach is validated by comparison with previous results reportedin the literature concerning a stationary cylinder (table 1), a flexibly mounted non-rotating cylinder (table 2) and a rigidly mounted rotating cylinder (figure 4). In thethree cases, the present results are in good agreement with previous work, whichensures the validity of the parametric study reported in this paper.
All the simulations are initialized with the established periodic flow past a stationarycylinder at Re= 100; then the forced rotation is started and the cylinder is released.The analysis is based on time series of more than 40 oscillation cycles (or vortexshedding cycles in the absence of vibrations), collected after the initial transient diesout. Convergence of each simulation is established by monitoring the time-averagedand r.m.s. values of the fluid force coefficients and body displacement.
3. Structural responsesThe responses of the flexibly mounted cylinder subjected to forced rotation are
quantified in this section. For periodic oscillations and forces, as those typically
–0.02
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(b)
3 4 52 3 4 52
(d)
3 4
Polynomial order52 3 4
Polynomial order52
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(c)
FIGURE 3. Time-averaged (a,c) in-line and (b,d) cross-flow force coefficients as functionsof the polynomial order, for (a,b) a rigidly mounted rotating cylinder at α= 4 and (c,d) aflexibly mounted rotating cylinder at (α,U?)= (3.75, 13) at Re= 100.
Study St Cx (Cy)max
Braza, Chassaing & Ha Minh (1986) 0.16 1.28 0.29Stansby & Slaouti (1993) 0.166 1.32 0.35Kang et al. (1999) 0.165 1.32 0.32Zhou, So & Lam (1999) 0.162 1.48 0.31Kim, Kim & Choi (2001) 0.165 1.33 0.32Shiels, Leonard & Roshko (2001) 0.167 1.33 0.30Stojkovic et al. (2002) 0.165 1.34 0.33Shen, Chan & Lin (2009) 0.166 1.38 0.33Present 0.164 1.32 0.32
TABLE 1. Strouhal number, time-averaged in-line force coefficient and maximum cross-flow force coefficient for a stationary cylinder (i.e. rigidly mounted and non-rotating), atRe= 100, from previous work and the current paper.
observed in this work, the time-averaged displacement of the cylinder can beexpressed as ζ =U?2Cy/8π2m by considering the time-averaged form of the structuredynamics (2.1). As shown in figure 4(b), where the time-averaged cross-flow forcecoefficient is plotted as a function of the rotation rate in the rigidly mounted cylinder
(a) (b)
1.0 2.0 3.00.5 1.5 2.5 3.5 1.0 2.0 3.00.5 1.5 2.5 3.5
0
1.0
1.2
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–0.2 Present studyStojkovi c et al. (2002, 2003)Kang et al. (1999)
0
–10
–15
–20
–25
–5
0.40 0.40
FIGURE 4. Time-averaged (a) in-line and (b) cross-flow force coefficients for a rigidlymounted rotating cylinder as functions of the rotation rate, at Re= 100. In panel (b) thedotted line indicates the cross-flow force coefficient in the potential flow case.
Study (ζ )max f Cx (Cy)max
Shiels et al. (2001) 0.58 0.196 2.22 0.77Shen et al. (2009) 0.57 0.190 2.15 0.83Present 0.57 0.188 2.08 0.88
TABLE 2. Maximum amplitude of vibration, vibration frequency, time-averaged in-lineforce coefficient and maximum cross-flow force coefficient for a flexibly mountednon-rotating (α= 0) cylinder at Re= 100, for m= 1.25, ξ = 0 and U?= 4.46, from previouswork and the current paper.
case, the magnitude of Cy increases with α (Magnus effect (Prandtl 1926)); thenegative sign is due to the anticlockwise rotation. As also noted in previous studies(e.g. Mittal & Kumar 2003), Cy differs substantially from the prediction throughpotential flow theory (Cy=−2πα; dotted line), and exceeds the limit of 4π proposedby Prandtl (1926). A similar trend of Cy versus α is noted in the flexibly mountedcylinder case, as shown in § 5. As a consequence, for a given reduced velocity, aglobal decrease of ζ is expected as α increases. This behaviour is verified in figure 5,where the time-averaged displacement of the body is plotted as a function of thereduced velocity, for each rotation rate. The actual responses of the flexibly mountedcylinder follow the trends obtained by considering the above time-averaged formof the dynamics equation and the mean force coefficients issued from the rigidlymounted body case (dashed lines). Some differences can be noted, as for instance forα = 3.75 around U? = 14; this highlights the alteration of the time-averaged forcesunder body vibrations, which is discussed in § 5.
The maximum amplitude of the cylinder oscillation about its time-averaged position(ζ ) is plotted as a function of the rotation rate and reduced velocity in figure 6(a,b).The cylinder exhibits large-amplitude vibrations over a wide zone of the parameterspace. As a general trend, it appears that the response peak amplitude and the widthof the U? range where large-amplitude oscillations occur first increase with α andthen drop abruptly. For a better visualization of the vibration region in the (α, U?)
–18
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–14
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–2
4 6 8 10 12 14 16 18 20 22 24 26 28 30 32–20
0
FIGURE 5. Time-averaged displacement as a function of the reduced velocity. For eachrotation rate, a dashed line indicates the displacement induced by the time-averaged cross-flow force issued from the rigidly mounted cylinder case, as plotted in figure 4(b).
domain, the area where oscillations larger than 0.05D are observed is represented bythe hatched zone in figure 6(c). Vibrations are noted for α < 1.8, in a region wherethe flow past a rigidly mounted cylinder is unsteady, but also for α > 1.8, where therotation suppresses the vortex formation in the rigidly mounted body case (e.g. Kanget al. 1999; Stojkovic et al. 2002). No vibration is observed for α = 4.
As shown in figure 7(a), where the maximum amplitude of oscillation is plotted asa function of U?, the cylinder may reach vibration amplitudes close to 1.9D underforced rotation, i.e. more than three times larger than the peak amplitude in thenon-rotating case. A drift of the peak amplitude towards higher values of U? canbe noted as α increases, for α 6 3; the trend reverses for larger values of α. Thepresent vibrations exhibit lower amplitudes than the galloping-like responses reportedby Stansby & Rainey (2001) and Yogeswaran & Mittal (2011) for a rotating cylinderfree to oscillate in both the in-line and cross-flow directions. The amplitude ofgalloping vibrations increases continuously with U?. Instead, the response amplitudesof the present rotating cylinder exhibit bell-shaped evolutions as functions of thereduced velocity, which resemble the typical VIV behaviour of the non-rotating case(α = 0). This suggests that, like VIV, the vibrations of the present rotating cylindermay occur under a wake–body synchronization mechanism, as opposed to gallopingvibrations, which do not involve such synchronization. This aspect is clarified in thefollowing sections.
The oscillations of the rotating cylinder are generally periodic and exhibit a strongsinusoidal nature. The spectral amplitudes of the higher harmonics that may appear inthe response always remain below 5 % of the first harmonic amplitude. The vibrationfrequency normalized by the natural frequency in vacuum (f ? = f /fn, with f the non-dimensional vibration frequency) is plotted as a function of the reduced velocity infigure 7(b). The overall evolution of the response frequency in the rotating cylindercases is comparable to the VIV behaviour observed at α = 0. In the range of low
05
1015 20
25 30 0
12
3
4
05
1015
2025
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01
23
4
Unsteady flow
Steady flow
Vibrations
(a)
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0
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2.0
FIGURE 6. (a,b) Maximum amplitude of vibration as a function of the rotation rateand reduced velocity. (c) Region of vibration (hatched area) in the rotation rate–reducedvelocity domain. The limit of the vibration region is indicated by the plain black line.In panel (c), the dashed line represents the limit between the steady and unsteady flowregimes, outside the vibration region.
reduced velocities, the vibration frequency bends downwards to follow approximatelythe normalized vortex shedding frequency observed in the stationary cylinder case(dashed line; Strouhal number in the range 0.163–0.165 in the rotating cases, thusclose to the non-rotating case value). At higher values of U?, the response frequencyremains relatively constant, but generally differs from the natural frequency (f ? 6= 1).The initial bending towards the Strouhal frequency, observed at low values of U?, alsoappears for rotation rates associated with a suppression of the vortex shedding in therigidly mounted cylinder case. A global decrease of the vibration frequency can benoted as α increases; this phenomenon will be connected to the variability of theeffective added mass in § 5.
(a)
(b)
0.2
0.4
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0.8
1.0
1.2
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1.8
4 6 8 10 12 14 16 18 20 22 24 26 28 30 32
4 6 8 10 12 14 16 18 20 22 24 26 28 30 32
0.92
0.94
0.96
0.98
1.00
0.90
0
2.0
0.88
1.02
FIGURE 7. (a) Maximum amplitude of vibration and (b) normalized vibration frequencyas functions of the reduced velocity. In panel (b), the dashed line indicates the normalizedfrequency of vortex shedding in the stationary cylinder case.
The flow patterns occurring in the wake of the freely vibrating rotating cylinder areexamined in the next section.
4. Wake patternsThe vortex shedding patterns encountered downstream of a non-rotating cylinder
subjected to forced or free vibrations have been identified and classified in previousstudies (Williamson & Roshko 1988; Brika & Laneville 1993; Govardhan &Williamson 2000; Carberry, Sheridan & Rockwell 2005). These works have shownthat the flow patterns may differ substantially from the alternate shedding observed inthe stationary cylinder case. Similarly, the impact of forced rotation on the wake of arigidly mounted cylinder has been extensively analysed (Badr et al. 1990; Stojkovicet al. 2002; Mittal & Kumar 2003; Pralits et al. 2010). The question that arises iswhether similar patterns will appear in the flow when the cylinder is subjected tofree vibrations under forced rotation. This question is addressed in this section, andparticular attention is paid to the possible occurrence of new wake patterns.
2S
U
P
0.5
1.0
1.5
2.0
2.5
3.0
3.5
6 8 10 12 14 16 18 20 22 24 26 28 30
C(2S)
0
4.0
324
(steady)
(steady)
FIGURE 8. Wake pattern as a function of the rotation rate and reduced velocity. The limitof the vibration region is indicated by the plain black line. The black dots denote the casesconsidered in figures 9–11.
A map of the different wake patterns in the (α,U?) domain is presented in figure 8.The map is based on two-dimensional flow simulation results. The three-dimensionaltransition that occurs at high rotation rates (α > 3.5) is investigated in § 6. Anumber of additional simulations have been performed to locate the limits of theregions associated with each wake pattern. However, it should be mentioned that thelimits indicated in figure 8 are approximate, since pattern switching and hysteresisphenomena may appear in the transition zones (Khalak & Williamson 1999; Prasanth& Mittal 2008). The different wake patterns are illustrated in figures 9–11 byinstantaneous isocontours of spanwise vorticity, for selected values of the pair (α,U?),which are designated by black dots in figure 8. In these figures, the cross-flowtrajectory of the body centre is indicated by the black segments.
The limit of the vibration region identified in § 3 is denoted by a plain black line infigure 8. Outside the vibration region, the wake patterns are those reported in previousstudies concerning rigidly mounted rotating cylinders. At low rotation rates, for α <1.8, the wake exhibits the 2S pattern, which is characterized by two counter-rotatingvortices shed per cycle, as in the stationary cylinder case (figure 9a). As also notedin previous studies (e.g. Mittal & Kumar 2003), the rotation induces an asymmetryin the strength of the positive and negative vortices, and an upward deviation of thewake. For α> 1.8, the wake is composed of two layers of vorticity of opposite signsand deflected upwards (figure 9b,c); the flow is steady. At sufficiently high rotationrates, the negative-vorticity tongue wraps around the cylinder and a switch of the twolayers of vorticity can be noted in the wake, i.e. the positive-vorticity layer is nowlocated above the negative-vorticity layer (figure 9c). In this case, the wake resemblesto some extent a jet-like wake. The inversion of the vorticity layers can be related tothe change observed in the sign of the in-line force (i.e. drag) at α≈ 3.7, as shown infigure 4(a). The two steady wake patterns are referred to as D+ and D−, in referenceto the positive or negative value of the drag. The switch from positive to negativedrag is chosen to define the limit between the D+ and D− pattern regions.
(a)
(b)
(c)
FIGURE 9. (Colour online) Instantaneous isocontours of vorticity: (a) 2S pattern, (α,U?)=(0.5,4), ωz=±0.4; (b) D+ pattern, (α,U?)= (2.5,4), ωz=±0.2; (c) D− pattern, (α,U?)=(3.75, 4), ωz = ±0.2. Positive and negative vorticity values are plotted in pale grey anddark grey (yellow and blue online), respectively. Part of the computational domain isshown.
The evolution of the structural response amplitude as a function of the reducedvelocity, presented in § 3 (figure 7a), suggested that the vibrations of the rotatingcylinder were synchronized with wake oscillations. As shown in § 5 by spectralanalysis, this hypothesis is verified: within the vibration region identified in the(α, U?) domain, the cylinder response frequency coincides with the predominantfrequency of the wake, i.e. the lock-in condition is established.
Within the vibration region, all the wake patterns encountered thus occur underwake–body synchronization. The area of lower rotation rates is characterized by the 2Spattern (figure 10a) and variations of the 2S pattern (figure 10b,c). In the range of lowreduced velocities, the vortices tend to coalesce in the wake (figure 10b); this pattern,also observed in a similar range of U? in previous studies concerning free vibrationsof a non-rotating cylinder (e.g. Singh & Mittal 2005), is referred to as C(2S). In therange of high reduced velocities, the two counter-rotating vortices shed during eachcycle regroup and form pairs, which resemble the P pattern reported by Williamson& Roshko (1988) for forced vibrations of large amplitude. At higher rotation rates, theregion of intermediate reduced velocities is dominated by a mixed pattern composed ofa pair (P) of vortices coupled to a single (S) vortex (figure 11a). This pattern, referredto as P+ S, has previously been observed in the wake of non-rotating cylinders under
(a)
(b)
(c)
P
FIGURE 10. (Colour online) Instantaneous isocontours of vorticity: (a) 2S pattern,(α, U?) = (1, 6.5), ωz = ±0.4; (b) C(2S) pattern, (α, U?) = (1, 5.5), ωz = ±0.4; (c) Ppattern, (α, U?)= (1.5, 10), ωz =±0.3. Positive and negative vorticity values are plottedin pale grey and dark grey (yellow and blue online), respectively. Part of the computationaldomain is shown. The trajectory of the cylinder centre is indicated by the black segment.
forced and free oscillations (Blackburn & Henderson 1999; Singh & Mittal 2005). Inthe range of intermediate reduced velocities and for α > 2.5 approximately, a fourthvortical structure appears during each shedding cycle, as shown in figure 11(b). Thistype of shedding clearly differs from the 2P pattern where two vortex pairs of similartopology are shed per cycle; as an extension of the P+ S pattern, this pattern may bereferred to as T+S, by regrouping the three vortices into a triplet (T), as indicated infigure 11(b). Shedding of two vortex triplets (2T) has been observed for a non-rotatingcylinder subjected to both in-line and cross-flow VIV (e.g. Dahl et al. 2007). To theauthors’ knowledge, the existence of the present T + S pattern has not previouslybeen reported. The last zone of the vibration region, principally associated with highreduced velocities, is dominated by another novel type of wake pattern, which was notdocumented in the literature on cylinder VIV: the wake is composed of two undulatingvorticity layers, without shedding of vortical structures (figure 11c). Because of itsundulatory nature, this pattern is referred to as U.
Previous studies concerning oscillating non-rotating cylinders have highlighted theinfluence of the oscillation amplitude and frequency on wake pattern selection (e.g.Williamson & Roshko 1988). In order to examine how the flow topology relates tothe vibration characteristics in the present rotating cylinder case, the pattern selected
(a)
P
S
S
T
(b)
(c)
FIGURE 11. (Colour online) Instantaneous isocontours of vorticity: (a) P + S pattern,(α,U?)= (2.5, 11), ωz =±0.2; (b) T+ S pattern, (α,U?)= (3.5, 13), ωz =±0.15; (c) Upattern, (α,U?)= (3, 25), ωz =±0.1. Positive and negative vorticity values are plotted inpale grey and dark grey (yellow and blue online), respectively. Part of the computationaldomain is shown. The trajectory of the cylinder centre is indicated by the black segment.
for each pair (α, U?) is indicated in figure 12 in the response amplitude–frequencydomain. Except in a small area in the low-frequency zone (f ≈ 0.05) where the Uand T+S patterns overlap, each wake pattern appears to be associated with a specificregion of the (ζmax, f ) domain. The 2S pattern and its variations (C(2S) and P) aremainly related to structural responses of moderate amplitudes and high to moderatefrequencies. Similarities may be noted with the map of wake patterns obtained throughforced vibration experiments by Williamson & Roshko (1988), in the non-rotatingcylinder case: the transition from 2S to C(2S) patterns as the vibration frequencyincreases, as well as the occurrence of the P+ S pattern for ζmax > 0.8 approximately,are in agreement with this previous work. Owing to the imposed rotation, significantdifferences exist, as for instance the occurrence of the P pattern in the region oflow-amplitude responses. The P + S and T + S patterns are associated with large-amplitude vibrations at moderate to low frequencies; the T + S pattern dominatesthe region of maximum-amplitude vibrations. It can be observed that the U pattern,which dominates the low-frequency zone, appears over a wide range of oscillationamplitudes.
As mentioned above, the vibrations of the rotating cylinder develop under thelock-in condition, which is the key mechanism of VIV. However, the self-sustained
1.0
1.8
1.6
1.4
1.2
0.8
0.6
0.4
0.2
0.04 0.06 0.08 0.12 0.14 0.16 0.180.10
f02.020.0
0
2.02SC(2S)P
U
FIGURE 12. Wake pattern as a function of the vibration maximum amplitude andfrequency.
oscillations occurring in the region of the U pattern, and thus excited by the flow inthe absence of vortex shedding, cannot be rigorously called vortex-induced vibrations.Therefore, the more general terminology flow-induced vibrations under wake–bodysynchronization is preferred to designate the responses of the rotating cylinder. Thisterminology also allows one to distinguish the present fluid–structure interactionphenomenon from galloping, which does not involve the lock-in condition.
The next section focuses on the forces exerted by the flow on the flexibly mountedrotating cylinder.
5. Fluid forcesIn order to further investigate the interaction mechanisms, a detailed analysis of
the fluid forces is reported in this section. An overview of the typical features ofthe fluid forces and energy transfer is presented in § 5.1. Systematic statistical andspectral analyses are performed in § 5.2 with an emphasis on the occurrence ofhigher harmonics in the force spectra. The phasing mechanisms between force anddisplacement are addressed in § 5.3. The effective added mass induced by the fluidis examined in § 5.4.
5.1. Temporal evolutions in selected configurationsThree configurations associated with three points in the (α,U?) domain are consideredat first to highlight some properties of the fluid forces. Focus is placed on thecross-flow force, since it is the component that drives the cylinder response. A globalanalysis encompassing the entire parameter space and including the in-line forcecomponent is reported in the following subsections. The selected configurations covera wide range of body oscillation amplitudes and frequencies; they correspond tothe 2S (figure 13), T + S (figure 14) and U (figure 15) wake patterns identified in§ 4. For each case, time series of the fluctuations of the cross-flow force coefficient(Cy) and of the pressure (Cp
y ) and viscous (Cvy) parts of this force coefficient (figures
0 1 2 3 4 5 6 7
0 1 2 3 4 5 6 73
0
0.1
–0.1
0
0.4
–0.4
0
0.4
–0.4
0
0.4
–0.4
0.2
–0.2
0
0.4
–0.4
0
0.4
–0.4
0.2
–0.2
Time
(d)
(e)
( f )
(g)
1 2 3 4 5 6 70
0.2
–0.2
0.8
–0.8
0.8
–0.8
0.8
–0.8
(a)
(b)
(c)
FIGURE 13. (Colour online) (a–c) Temporal evolution of the cross-flow force coefficientfluctuation (dashed line; left axis) and cylinder displacement fluctuation (solid line; rightaxis) for (α, U?) = (1, 6.5) – 2S pattern: (a) total, (b) pressure and (c) viscous. Theinstantaneous power due to the corresponding force component is also shown (dash-dottedline; left axis). (d–g) Instantaneous isocontours of vorticity at the times indicated byvertical dotted lines in panels (a–c). Positive and negative vorticity values are plotted inpale grey and dark grey (yellow and blue online), respectively, in the range ωz=±2.5. Partof the computational domain is shown. The trajectory of the cylinder centre is indicatedby the black segment.
–2
–1
0
1
2
(b)
0 5 10 15
0 5 10 15
5 10
–2
–1
0
1
2
–2
–1
0
1
2
0
1.25
–1.25
2.50
–2.50
0
0.75
–0.75
1.50
–1.50
(c)
510
Time
–2
–1
0
1
2
(a)
(e)
( f )
(g)
(d)
FIGURE 14. (Colour online) Same as figure 13, but now for (α,U?)= (3.5, 13) – T+ Spattern. In panels (d–g), the isocontours of vorticity are in the range ωz =±1.
13–15, panels (a–c), respectively; left axis) are plotted over one period of the cylinderoscillation. The fluctuation of the cylinder displacement is also plotted in these figures(right axis). In figures 13–15, panels (d–g), visualizations of the near-wake regionthrough isocontours of vorticity are presented at four time instants indicated by blackdotted lines in panels (a–c).
In the three cases, it can be observed that the unsteady wake pattern and thecylinder oscillation are synchronized, i.e. the lock-in condition is established. The
0
0.1
–0.1
0.2
–0.2
(a)
–1.0
–0.5
0
0.5
1.0
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–1.0
1.0
0 5 10 15 20 25
0 5 10 15 20 25
0 5 10 15 20 25
0
0.15
–0.15
0.30
–0.30
(b)
0
0.250
–0.250
0.125
–0.125
(c)
Time
0
0.50
–0.50
1.00
–1.00
(e)
( f )
(g)
(d)
FIGURE 15. (Colour online) Same as figure 13, but now for (α,U?)= (3, 25) – U pattern.In panels (d–g), the isocontours of vorticity are in the range ωz =±1.
magnitudes of the cross-flow force and its pressure and viscous parts exhibit strongvariations from one configuration to another. However, some features persist for thethree cases. In all cases, the body displacement is close to sinusoidal, which is notthe case for the force coefficient; this is related to the occurrence of large higherharmonics, as will be discussed in § 5.2. As expected in the presence of forcedrotation, which causes an asymmetry of the flow, the symmetry of Cy over eachhalf-cycle of oscillation is broken. The asymmetry is mainly due to the pressure
part of the force, whereas the viscous part remains relatively symmetrical. A jointmonitoring of the time series and flow snapshots allows one to relate some eventsin the evolution of Cp
y to the wake unsteadiness. For instance, the formation ofa negative-vorticity vortex in figure 13(e) coincides with a positive peak of Cp
y(figure 13b), i.e. the vortex seems to ‘pull’ the cylinder upwards. Conversely, theformation of a positive-vorticity vortex in figure 13(d,g) seems to ‘pull’ the cylinderdownwards and may be connected with a negative peak of Cp
y . However, it shouldbe recalled that vortex shedding is not required to observe fluctuations in the fluidforce and associated structural vibrations, as illustrated in figure 15 (U wake pattern).Among the three selected configurations, differences may be noted in the phasingbetween the body response and the fluid force, as for instance for Cp
y (figures 13band 14b). The force–displacement phasing mechanisms are addressed in § 5.3.
The instantaneous energy transfer between the flow and the structure is quantified bymeans of the fluid force coefficient in phase with the cylinder velocity. As previouslymentioned, the time-averaged component of the cross-flow force induced by therotation (Magnus effect) causes a deflection of the cylinder time-averaged position(figure 5). Here, attention is paid to the body oscillations about their time-averagedposition and thus to the energy transfer associated with the fluctuations of the forceand its pressure and viscous parts. The corresponding coefficients are defined asfollows:
Cyv =√
2Cyζ√ζ 2
, Cpyv =√
2Cpy ζ√ζ 2
, Cvyv =√
2Cvy ζ√ζ 2
. (5.1)
Positive values of these coefficients indicate that the flow excites the structuralvibrations; negative values indicate that the vibrations are damped by the flow.Time series of Cyv, Cp
yv and Cvyv are plotted in figures 13–15(a–c) (left axis). For
self-sustained periodic oscillations, such as those generally observed in the presentwork, the time-averaged net energy transfer is zero (Cyv = 0). Some observations,which are verified over the entire vibration region, can be made on the basis of theselected time series. The viscous part of the force principally damps the cylinderoscillations (C
vyv < 0), except in short time intervals where Cv
yv may become slightlypositive, as shown in figure 13(b). The excitation of the body by the flow is mainlydriven by the pressure part of the force (C
pyv > 0). Across the vibration region,
the shape of the energy transfer over one cycle of oscillation exhibits substantialvariations, which relate to the force frequency content and phasing; these aspects areinvestigated in the following subsections.
5.2. Statistics and spectral analysisThe time-averaged in-line and cross-flow force coefficients are plotted as functionsof the reduced velocity in figure 16. For each rotation rate, the values of Cx andCy in the rigidly mounted cylinder case are indicated by dashed lines. At a givenreduced velocity, the time-averaged force coefficients globally decrease as α increases.The decrease of Cy results in an increasing downward deviation of the cylinder time-averaged position, as reported in § 3. Significant modulations of Cx and Cy are notedwhen the cylinder vibrates.
In the in-line direction, the body oscillations are accompanied by an amplificationof Cx (figure 16a), as also noted in previous studies concerning non-rotatingcylinders (Khalak & Williamson 1999; Carberry et al. 2005; Bourguet, Karniadakis &
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
0
2.0
4 6 8 10 12 14 16 18 20 22 24 26 28 30 32
4 6 8 10 12 14 16 18 20 22 24 26 28 30 32
–16
–18
–14
–12
–10
–8
–6
–4
–2
0
(a)
(b)
FIGURE 16. Time-averaged (a) in-line and (b) cross-flow force coefficients as functions ofthe reduced velocity. The dashed line indicates, for each rotation rate, the time-averagedforce coefficient in the rigidly mounted cylinder case.
Triantafyllou 2011b). Regardless of the value of the rotation rate, Cx appears to followa similar trend within the vibration region: for each α, the largest amplification occursat the lowest reduced velocity, and then Cx decreases regularly across the vibrationwindow. This trend was previously observed for non-rotating cylinders (e.g. Khalak &Williamson 1999) and the present results show that it persists in the rotating cylindercase. In addition, this behaviour seems to be independent of the wake pattern, as itis observed across the entire vibration region.
In the cross-flow direction, both positive and negative deviations of Cy fromthe rigidly mounted body value may occur when the rotating cylinder oscillates(figure 16b). Comparison with the map of the wake patterns (figure 8) suggests apossible link between the sign of the deviation and the type of flow topology. To
0
0.2
–0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.0 1.81.61.41.20.80.60.40.2
–0.4
1.6
0.20
2S
P
U
C(2S)
FIGURE 17. Deviation of the time-averaged cross-flow force coefficient from the value inthe rigidly mounted cylinder case as a function of the maximum amplitude of vibration.The symbols represent the wake pattern.
clarify this aspect, the deviation of Cy is plotted in figure 17 as a function of themaximum vibration amplitude for all the points located within the vibration region,and the corresponding wake pattern is indicated by a symbol. As a general trend, itappears that the 2S pattern and its variations (P and C(2S)) are principally associatedwith negative deviations of Cy, whereas the other patterns (P+ S, T+ S and U) aremainly related to positive deviations. A second trend can be identified: in the rangeof positive deviations, the deviation magnitude appears to increase globally with theamplitude of vibration; such behaviour is not observed for negative deviations.
Previous work has shown that the amplitudes of fluctuation of the fluid forcesapplied on a non-rotating cylinder tend to increase with the amplitude of vibration(Bishop & Hassan 1964; Khalak & Williamson 1999). A comparable trend is observedin the rotating cylinder case, as illustrated in figure 18, where the r.m.s. value ofthe in-line force coefficient fluctuation (Cx) is plotted as a function of the maximumvibration amplitude. Some modulations are, however, noted, since the largest r.m.s.value of Cx for a given rotation rate does not necessarily coincide with the largestvibration amplitude. Similar observations can be made for the cross-flow forcecoefficient.
The r.m.s. values of the fluctuations of the cross-flow force coefficient and of itspressure and viscous parts are plotted as functions of the reduced velocity in figure 19,for all the studied rotation rates at which free vibrations develop, i.e. α 6 3.75. Foreach α, large variations of the r.m.s. values are noted within the vibration region. Ther.m.s. value of Cy exhibits a peak near the low-U? limit of the vibration window andthen decreases until the high-U? limit of the vibration window; such behaviour waspreviously identified in the non-rotating cylinder case (Khalak & Williamson 1999;Govardhan & Williamson 2000) and the imposed rotation has no significant impacton the overall trend. The pressure part closely follows the evolution of the total force,while the viscous part presents a smoother behaviour and generally lower r.m.s. values.
1.0 1.81.61.41.20.80.60.40.2 0.20
1
2
3
4
5
6
7
8
9
FIGURE 18. The r.m.s. value of the in-line force coefficient fluctuation as a function ofthe maximum amplitude of vibration.
The phasing of the cross-flow force and its pressure and viscous parts with the bodydisplacement is examined in § 5.3.
A spectral analysis is performed in the following in order to shed light on somefluid–structure interaction mechanisms and to emphasize the impact of the imposedrotation on the force frequency content. In figures 20 and 21, the power spectraldensities (PSDs) of the in-line and cross-flow force coefficients are plotted, for eachrotation rate, as functions of the reduced velocity. The PSDs are normalized by themagnitude of the largest peak. At each reduced velocity, the predominant frequency ofthe wake is indicated by a triangle; this frequency is based on the PSD of time seriesof the cross-flow component of the flow velocity at a point located 10D downstreamof the cylinder. The vortex shedding frequency in the rigidly mounted cylinder case isdenoted by a dashed line (α < 1.8). Within the vibration region, the body oscillationfrequency is indicated by a circle.
The cylinder vibration frequency and the wake frequency generally coincide. Thesynchronization between the cylinder oscillation and the flow unsteadiness wasillustrated in selected configurations in figures 13–15. The spectral analysis confirmsthat the flow-induced vibrations of the rotating cylinder occur under the lock-incondition, like VIV. However, as previously mentioned, the excitation of the rotatingcylinder by the unsteady flow does not necessarily involve vortex shedding; indeed,large-amplitude oscillations are also noted when the wake exhibits the U pattern.The present flow-induced vibrations under wake–body synchronization differ from thegalloping responses reported by Stansby & Rainey (2001) and Yogeswaran & Mittal(2011) for a rotating cylinder free to oscillate in both the in-line and cross-flowdirections, since the galloping vibrations are not synchronized with the wake.
Within the vibration region, the fluid forces only peak at the vibration frequency orat frequencies corresponding to higher harmonics of the vibration frequency; therefore,the fluid forces are synchronized with the structural response. This contrasts with theabove-mentioned galloping responses, where a high-frequency fluctuation of the forces,
4 6 8
(a)
5 10 15
10 20 30
0
0.2
0.4
0.6
0.8
1.0 (b () c)
0
0.2
0.4
0.6
0.8
1.0
r.m.s
.
(d)
r.m.s
.
0
0.5
1.0
1.5
2.0
(g () h)
r.m.s
.
0
0.5
1.0
1.5
2.0
10 20 300
0.5
1.0
1.5
(e)
0
0.5
1.0
1.5
5 10 15 20
4 6 8 10 4 6 8 100
0.5
1.0
1.5
( f )
0
0.5
1.0
1.5
5 10 15 20
5 10 15 20
(i)
0
0.2
0.4
0.6
0.8
1.0
FIGURE 19. (Colour online) The r.m.s. values of the total (Cy), pressure (Cpy ) and viscous
(Cvy) cross-flow force coefficient fluctuations as functions of the reduced velocity for
(a) α= 0, (b) α= 0.5, (c) α= 1, (d) α= 1.5, (e) α= 2, (f ) α= 2.5, (g) α= 3, (h) α= 3.5and (i) α = 3.75.
due to vortex shedding, is superimposed onto the low-frequency component caused bybody oscillation. The observed decreasing trend of the force frequencies within thevibration region thus corresponds to the decrease of the vibration frequency, whichis driven by the decrease of the natural frequency as U? increases. It is recalled thatU? = 1/fn and that the vibration frequency differs but remains relatively close to fn(figure 7b).
In the absence of rotation, the anti-symmetric nature of the vortex shedding patternresults in a ratio of 2 between the fundamental frequencies of the in-line and cross-flow forces. As previously noted in the rigidly mounted cylinder case (Mittal & Kumar2003), the symmetry breaking induced by the imposed rotation results in a switch toa frequency ratio of 1, i.e. the fundamental frequencies of Cx and Cy are the same.This phenomenon is also observed in the present case, outside the vibration region, asshown in figures 20 and 21. A slight second harmonic contribution may persist in theCx spectrum for α > 0 (figure 20b) but it tends to vanish as α increases. The impactof the imposed rotation on the force spectra within the vibration region is addressedin this subsection.
15 20 10 15 20 25 30 14 15
Freq
uenc
y
(g)
0.1
0.2
0.3
0.4
6 8 10 12 14 160.05
0.10
0.15
0.20
0.25
0.30
10 15 20
0.05
0.10
0.15
0.20
0.25
0.05
0.10
0.15
0.20
0.05
0.10
0.15
0.20
0.25
10 25 13 16
( f)
( i)
(e)
(h)
4 6 8
0.2
0.3
0.4
0.1
0.5Fr
eque
ncy
(a)
4 6 8 10
0.2
0.3
0.4
0.1
0.5 (c)(b)
4 6 8 10
0.2
0.3
0.4
0.1
0.5
5 10 15
Freq
uenc
y
(d)
0.1
0.2
0.3
0.4
FIGURE 20. (Colour online) PSD of the in-line force coefficient (Cx) as a function of thereduced velocity for (a) α= 0, (b) α= 0.5, (c) α= 1, (d) α= 1.5, (e) α= 2, (f ) α= 2.5,(g) α= 3, (h) α= 3.5 and (i) α= 3.75. For each reduced velocity, the PSD is normalizedby the magnitude of the largest peak. The greyscale levels range from 0 (white) to 1(black). The cylinder vibration frequency is indicated by a circle and the predominantfrequency of the wake by a triangle. In panels (a–d) (α < 1.8), the dashed line denotesthe vortex shedding frequency in the rigidly mounted cylinder case.
A striking feature of the force spectral content is the presence of large higherharmonics in the cross-flow direction. For instance, substantial contributions of thethird harmonic can be noted in the Cy spectrum at α = 0 (figure 21a), while boththe second and third harmonics exhibit significant peaks at α = 1.5 (figure 21d). Inthe non-rotating case, previous studies concerning flexible and flexibly mounted rigidcylinders have emphasized the existence of such higher harmonics and particularlythe possible occurrence of a large third harmonic component in the cross-flow forcespectrum (Jauvtis & Williamson 2004; Dahl et al. 2007; Vandiver, Jaiswal & Jhingran2009; Dahl et al. 2010). Modarres-Sadeghi et al. (2010) have highlighted the influenceof the force higher harmonics on the fatigue life of long marine risers. Wang, So& Chan (2003) and Wu, Ge & Hong (2012) proposed a simple model to relate theoccurrence of the force higher harmonics to the body displacement, for a non-rotatingcylinder subjected to a sinusoidal motion. Here, this model is extended to take intoaccount the symmetry breaking caused by the imposed rotation.
4 6 8 4 6 8 10
5 10 15 10 15 20
15 20 10 15 20 25 30 13 14 15 16
0.2
0.3
0.4
0.1
0.5
Freq
uenc
y(a)
Freq
uenc
y
(d)
Freq
uenc
y
(g)
(b)
(e)
(h)
(c)
(f )
(i)
0.1
0.2
0.3
0.4
0.5
4 6 8 100.1
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6 8 10 12 14 16
0.1
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0.25
0.05
0.10
0.15
0.20
0.05
0.10
0.15
0.20
0.25
10 25
FIGURE 21. (Colour online) Same as figure 20 for the cross-flow force coefficient (Cy)at the same values of α.
Assuming a periodic structural response of fundamental frequency f, the cylinderdisplacement can be expanded as
ζ =∞∑
n=0
ζn sin(2πnft+ φn), (5.2)
where ζn and φn denote the spectral amplitude and phase lag associated with the nthharmonic. Owing to the vibration, the instantaneous velocity vector of the cylinderis not aligned with the x axis. The angle between the x axis and the axis parallelto the instantaneous velocity vector is θ = arctan(−ζ ). The force coefficient parallelto the instantaneous velocity vector is referred to as Cd and the perpendicular forcecoefficient as Cl. The force coefficients in the stationary frame can be expressed asfollows:
Cx=Cd cos(θ)−Cl sin(θ), (5.3a)Cy=Cd sin(θ)+Cl cos(θ). (5.3b)
To simplify the analysis, θ is assumed to be small, which leads to
Cx=Cd +Clζ , (5.4a)Cy=−Cdζ +Cl. (5.4b)
The force coefficients expressed in the frame aligned with the instantaneous velocityvector are assumed to have a form analogous to the rigidly mounted cylinder case inthe unsteady flow regime:
Cd=Cd +Cd1 sin(2πft+ φd1)+Cd2 sin(4πft+ φd2), (5.5a)Cl=Cl +Cl1 sin(2πft+ φl1). (5.5b)
Here Cd and Cl are the time-averaged values of Cd and Cl; Cd1, Cl1 and Cd2 are thefirst and second harmonic amplitudes; and φd1, φl1 and φd2 are the associated phaselags. The first harmonic of Cd and the time-averaged value of Cl are introduced totake into account the symmetry breaking caused by the rotation; in the absence ofrotation, Cl = Cd1 = 0. The in-line and cross-flow force coefficients can be expressedas follows:
Cx=Cd +πf∞∑
n=1
nζn2Cl cos(2πnft+ φn)+Cl1 sin[2π(n+ 1)ft+ φl1 + φn]
−Cl1 sin[2π(n− 1)ft− φl1 + φn], (5.6)
Cy=Cl −πf∞∑
n=1
nζn2Cd cos(2πnft+ φn)+Cd1 sin[2π(n+ 1)ft+ φd1 + φn]
−Cd1 sin[2π(n− 1)ft− φd1 + φn] +Cd2 sin[2π(n+ 2)ft+ φd2 + φn]−Cd2 sin[2π(n− 2)ft− φd2 + φn]. (5.7)
Therefore, several higher harmonic components may arise in the Cx and Cy spectra,depending among other things on the spectral content of the structural response. Aspreviously mentioned, the cylinder oscillations remain close to sinusoidal. In the caseof pure sinusoidal vibrations, i.e. ζn = 0 for n > 1, the force coefficients become(without loss of generality φ1 is set equal to 0):
Cx=Cd +πf ζ1Cl1 sin(φl1)+Cd1 sin(2πft+ φd1)+ 2πf ζ1Cl cos(2πft)+Cd2 sin(4πft+ φd2)+πf ζ1Cl1 sin(4πft+ φl1), (5.8)
Cy=Cl −πf ζ1Cd1 sin(φd1)+Cl1 sin(2πft+ φl1)− 2πf ζ1Cd cos(2πft)−πf ζ1Cd2 sin(2πft+ φd2)−πf ζ1Cd1 sin(4πft+ φd1)
−πf ζ1Cd2 sin(6πft+ φd2). (5.9)
In the absence of rotation (Cl = Cd1 = 0), the above simple model indicates that,owing to the body oscillation, a third harmonic contribution may appear in the Cyspectrum, while the Cx spectrum is dominated by a single frequency equal to twicethe cross-flow fundamental frequency. Under forced rotation (Cl 6= 0 and Cd1 6= 0), themodel suggests that both second and third harmonic contributions may arise in theCy spectrum. The magnitude of the second harmonic of the in-line force in the rigidlymounted cylinder case (Cd2) decreases as α increases: therefore, the contribution of thethird harmonic in the Cy spectrum is expected to decrease as α increases. Owing to thelarge magnitude of the time-averaged cross-flow force in the rigidly mounted cylindercase (Cl), the first harmonic is expected to dominate the Cx spectrum, especially athigh rotation rates. The trends predicted by the model are confirmed by the simulationresults presented in figures 20 and 21. It is recalled that the above model, which is anextension of the model proposed by Wang et al. (2003), relies on several assumptions.
It is used in the present study only to illustrate the possible effect of the combinedoscillation and rotation of the cylinder on the force frequency content.
The principal results of the spectral analysis can be summarized as follows.The flow-induced vibrations of the rotating cylinder occur under wake–bodysynchronization and, within the vibration region, the fluid forces are synchronizedwith the wake–body oscillation. The symmetry breaking caused by the rotationresults in a switch of the ratio between the in-line and cross-flow force fundamentalfrequencies, from 2 for α = 0 to 1 for α > 0. Both within and outside the vibrationregion, the in-line force spectrum is dominated by the first harmonic contribution.In the absence of vibration, the cross-flow force spectrum presents a single firstharmonic component. In contrast, large higher harmonic components appear in thecross-flow force spectrum when the cylinder vibrates: for α = 0, as also noted inprevious works, a significant third harmonic contribution may be observed; for α > 0,both second and third harmonic components arise and the third harmonic contributiondecreases with increasing rotation rate.
For a given rotation rate, the contributions of each harmonic component to theoverall cross-flow force significantly vary across the vibration region. In particular, thefirst harmonic contribution seems to vanish for some values of the reduced velocity,for instance near (α,U?)= (1, 7) or (2, 12), as shown in figure 21. This phenomenonis related to the mechanism of phasing between the force and the body displacement,as discussed in the following subsection.
5.3. Phasing between force and displacementThe phasing between the fluid forcing and the body response has been thoroughlystudied in the literature related to cylinder VIV (Bearman 1984; Khalak & Williamson1999; Carberry et al. 2001; Williamson & Govardhan 2004; Leontini et al. 2006;Dahl et al. 2010). The force–displacement phasing of the freely vibrating cylinderunder forced rotation is investigated in this subsection. To present the typicalphasing mechanism, periodic structural response and cross-flow force coefficientof fundamental frequency f are considered. The structural response is defined as in(5.2) and the force coefficient is expanded in a similar way as
Cy =∞∑
n=0
Cyn sin(2πnft+ φyn), (5.10)
where Cyn and φyn designate the spectral amplitude and phase lag associated with thenth harmonic. Since the body oscillation is generally close to sinusoidal, particularattention is paid to the phasing between the first harmonic components of thedisplacement and the force. Without loss of generality, φ1 is selected equal to 0, sothat the phase difference between the first harmonics is equal to φy1. If ζ and Cyverify the dynamics equation (2.1), in which the structural damping is set to zero(ξ = 0), then (for f 6= fn)
φy1 = 0 or 180 (5.11)
and4π2f 2
n (1− f ?2)ζ1 = Cy1
2mcos(φy1), (5.12)
where, as previously defined, f ? = f /fn. As a result, the system may exhibit twopossible phasing states: (i) the first harmonics of the force and displacement are
5 6 7 8 6 8 10
10 15 20
15 20 10 15 20 25 30 13 14 15 16
90
180
270
Phas
e di
ffer
ence
(de
g.)
0
360
(a)
90
180
270
Phas
e di
ffer
ence
(de
g.)
0
360
90
180
270
0
360
(b)
90
180
270
0
360
(c)
5 6 7 8
90
180
270
0
360
90
180
270
0
360
6 8 10 12 14 166 8 10 12
(d)
Phas
e di
ffer
ence
(de
g.)
(g)
( f )(e)
90
180
270
0
360
(h)
90
180
270
0
360
(i)
90
180
270
0
360
10 25
FIGURE 22. (Colour online) Phase differences between the total (Cy), pressure (Cpy ) and
viscous (Cvy) cross-flow force coefficients and the cylinder displacement as functions of the
reduced velocity for (a) α= 0, (b) α= 0.5, (c) α= 1, (d) α= 1.5, (e) α= 2, (f ) α= 2.5,(g) α = 3, (h) α = 3.5 and (i) α = 3.75.
in phase (φy1 = 0) and the vibration frequency is lower than the natural frequencyin vacuum (f ? < 1); (ii) the first harmonics of the force and displacement are inantiphase (φy1= 180) and the vibration frequency is larger than the natural frequencyin vacuum (f ? > 1). A phase jump occurs when the vibration frequency passesthrough the value of the natural frequency; for f ? = 1, the contribution of the firstharmonic of the force (Cy1) vanishes since Cy1 cos(φy1) = Cy1 sin(φy1) = 0. Previouswork concerning VIV of non-rotating cylinders has emphasized the existence of thetwo above phasing states, over the lock-in range. It should be mentioned that (5.11)and (5.12) hold in both the non-rotating and rotating cylinder cases.
In order to clarify the impact of the cylinder rotation on the force–displacementphasing mechanism, the phase difference between the first harmonics of ζ and Cy(φy1)
is plotted as a function of the reduced velocity in figure 22, for each rotation rate(squares). For α 6 2, both phasing states (φy1 = 0 and 180) are observed withinthe vibration window, whereas the force and displacement remain in phase at higherrotation rates, regardless of the value of U?. Comparison with the evolution of thevibration frequency (figure 7b) confirms that the phase jump coincides rigorously
with f ? = 1, i.e. the cylinder vibrates at the natural frequency. As reported in § 3,the vibration frequency exhibits a global decrease as the rotation rate increases; thisexplains why the phase jump tends to drift towards higher values of the reducedvelocity as α increases. For α > 2.5 cases, the vibration frequency is always lowerthan the natural frequency: in these cases, no phase jump occurs and only the type (i)phasing state is observed.
As expected on the basis of the above analysis, low magnitudes of the first harmoniccomponent can be noted near the phase jumps in figure 21. In these regions, thecontributions of the higher harmonic components become predominant, as for instancenear U? = 7 at α = 1.
As also noted in previous studies concerning non-rotating cylinders (e.g. Govardhan& Williamson 2000), the phase jump does not coincide with a switch of the wakepattern. In order to shed some light on the physical nature of the phase discontinuity,the phase differences between the first harmonic of ζ and the first harmonics of thepressure (Cp
y ) and viscous (Cvy) parts of Cy are also plotted in figure 22 as circles and
triangles, respectively. Contrary to the total cross-flow force, the phase differencesassociated with Cp
y and Cvy, referred to as φp
y1 and φvy1, respectively, vary continuously
as functions of U?. Hence, the phase jump of Cy is not due to a jump in the phaseof its pressure or viscous parts; instead, it results from the combination of twoforce components whose phases vary smoothly across the vibration window. Thephase difference associated with the viscous part remains in a narrow range of phasedifference angles, between 30 and 90 approximately; a slight overall increase maybe noted as a function of α, while for each rotation rate, φv
y1 tends to increase withU?. The phase difference associated with the pressure part exhibits larger variations,especially in the range of low reduced velocities. As indicated in § 5.1, some eventsin the evolution of the pressure part of Cy can be connected to the formation ofvortices in the wake. Therefore, it is expected that the phasing of vortex formationwill influence the phasing of Cp
y . However, it can be observed that φpy1 may still vary
in the absence of vortex shedding, as for instance for U? > 22 at α = 3.5, where thewake is characterized by the U pattern.
The fact that no phase jump of the cross-flow force occurs at high rotation ratewas related to the global decrease of the vibration frequency as α increases. In thefollowing, this trend of the structural response frequency is investigated in light of thevariability of the effective added mass.
5.4. Effective added massPrevious studies concerning cylinder VIV have shown that the effective added masscoefficient induced by the fluid force in phase with the body acceleration maysubstantially depart from the potential flow value of 1 (Sarpkaya 1979; Vikestad,Vandiver & Larsen 2000; Dahl et al. 2010; Bourguet, Karniadakis & Triantafyllou2011a). The effective added mass coefficient can be defined as follows:
Cm =− 2π
Cyζ
ζ 2. (5.13)
For periodic displacement (5.2) and force coefficient (5.10), an effective added masscoefficient, involving the response first harmonic component only, may be defined byremoving all the higher harmonic contributions in (5.13) (Dahl et al. 2010):
C1m =
Cy1 cos(φy1)
2π3f 2ζ1. (5.14)
0
0.2
–0.2
0.4
–0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
4 6 8 10 12 14 16 18 20 22 24 26 28 30 32
FIGURE 23. (Colour online) Effective added mass coefficient associated with the responsefirst harmonic component (C1
m defined in (5.14)) as a function of the reduced velocity.The effective added mass coefficient based on the total response (Cm defined in (5.13)) isindicated by a dot (red online) for each pair (α,U?).
In the case of sinusoidal responses, C1m is equal to Cm. The coefficient C1
m issignificant as it relates directly to the vibration frequency as follows (Williamson &Govardhan 2004):
f ? =√
mm+ 1
4πC1m
. (5.15)
For each rotation rate, C1m is plotted as a function of the reduced velocity in
figure 23. The values of Cm are also presented, for comparison purposes (dots; redonline).
Slight differences may be noted between Cm and C1m, especially at high rotation
rates, due to the occurrence of small higher harmonic components in the response.However, as previously mentioned, the higher harmonic contributions are limitedand the response remains close to sinusoidal in all cases, which explains the globalsimilarity between Cm and C1
m evolutions. The effective added mass coefficientgenerally differs from 1. The decreasing trend observed as a function of the reducedvelocity in the non-rotating cylinder case was also reported in previous studies (e.g.Dahl et al. 2010). The global increase of the effective added mass coefficient asα increases is associated with the previously mentioned decrease of the vibrationfrequency. For α > 2.5 cases, the vibration frequency remains lower than the naturalfrequency (figure 7b) and the cross-flow force phase jump vanishes (figure 22). Thisphenomenon is related to the strictly positive value of the effective added mass overthe entire vibration window.
So far, the analysis of the coupled fluid–structure system has been carried out on thebasis of two-dimensional simulations of the flow. The three-dimensional transition inthe flow at high rotation rates and its possible influence on the interaction mechanismsare investigated in the next section.
6. Three-dimensional transitionPrevious studies concerning rigidly mounted rotating cylinders have shown that
rotation substantially alters the flow three-dimensional transition scenario comparedto the non-rotating case (Mittal 2004; El Akoury et al. 2008; Meena et al. 2011; Raoet al. 2013b). At Re= 100, the recent linear stability analyses reported by Pralits et al.(2013) and Rao et al. (2013a) predict that the flow should become three-dimensionalfor α ≈ 3.7. The occurrence of the three-dimensional transition under the effectof forced rotation is examined in the following on the basis of three-dimensionaldirect numerical simulation results. The objective here is not to provide an extensiveanalysis of the transition mechanisms; instead, this section aims at clarifying thepotential impact of the three-dimensional transition on the fluid–structure interactionphenomena previously described under the two-dimensional flow assumption. Thecase of a rigidly mounted cylinder is considered as a first step in § 6.1 in order toidentify the region of transition and quantify its influence on the fluid forces. Theflexibly mounted cylinder case is addressed in § 6.2.
6.1. Rigidly mounted cylinderThe flow past a rigidly mounted cylinder subjected to forced rotation is visualizedin figure 24 by means of instantaneous isosurfaces of the spanwise and streamwisevorticity components, for three values of the rotation rate, α ∈ 3.5, 3.75, 4.0. Theflow is steady in this range of rotation rates, as also noted in the two-dimensionalsimulations. The present three-dimensional simulations show that the flow remainstwo-dimensional at α= 3.5. In contrast, an undulation of the spanwise vorticity layersis observed at α=3.75 and tends to amplify at α=4.0; this undulation is accompaniedby the development of elongated vortical structures in the streamwise direction. As aresult, the flow past a rigidly mounted rotating cylinder at Re = 100 becomes three-dimensional at a critical rotation rate located between 3.5 and 3.75; this observation isin agreement with the predictions based on linear stability analyses (Pralits et al. 2013;Rao et al. 2013a). The three-dimensional flow pattern is comparable to those reportedby Mittal (2004) and Meena et al. (2011) at Re=200. The spanwise wavelength of thethree-dimensional pattern is approximately equal to 1.7D; the above-mentioned linearstability analyses predict a similar wavelength.
In order to quantify the impact of the three-dimensional transition on the fluidforces, the evolutions of Cx and Cy are plotted along the cylinder span in figure 25,for α=3.75 and α=4.0. It is recalled that the flow, though three-dimensional, remainssteady for both rotation rates, thus Cx = Cx and Cy = Cy. Spanwise fluctuations ofthe force coefficients can be noted in both directions, with an amplification of thefluctuations as α increases; this amplification correlates to the previously mentionedincreasing spanwise undulation of the flow. The span-averaged values of the forcecoefficients are indicated in figure 25 by dashed lines while the values issued fromtwo-dimensional simulations are denoted by dash-dotted lines. In the range of rotationrates considered in the present work (α 6 4.0), it appears that the three-dimensionaltransition in the flow has only a very limited influence on the span-averaged fluidforces.
Two steady flow patterns, the D+ and D− patterns, have been identified outside ofthe vibration region, on the basis of two-dimensional simulations in § 4. The presentthree-dimensional simulations allow one to refine the description of these patterns. Thelimit between the D+ and D− flow patterns was defined based on a change in thesign of the in-line force. As can be noted in figure 25(a), the passage from positiveto negative values of span-averaged Cx occurs in the range α ∈ [3.5, 3.75] (as in
x
y
zx
y
z
x
y
z
x
y
z
x
y
z
x
y
z
(a () b)
(c () d)
(e () f )
FIGURE 24. (Colour online) Instantaneous isosurfaces of (a,c,e) spanwise (ωz=±0.2) and(b,d,f ) streamwise (ωx=±0.2) vorticity in the rigidly mounted cylinder case, for (a,b) α=3.5, (c,d) α = 3.75 and (e, f ) α = 4.0. Part of the computational domain is shown.
the two-dimensional case), which is also the range associated with the onset of thethree-dimensional transition. As a result, the D+ pattern is essentially two-dimensional,whereas the D− pattern is characterized by an undulation of the flow along the span,in addition to the inversion of the spanwise vorticity layers reported in § 4 and alsovisible in figure 24(c,e).
The three-dimensional transition occurs in a range of rotation rates that correspondsto the upper limit of the vibration region determined in § 3, on the basis oftwo-dimensional simulations. This is also the area where the largest vibrationamplitudes have been observed. As a consequence, the effect of the three-dimensionaltransition on the behaviour of the coupled fluid–structure system needs to be clarifiedin order to ensure the validity of the previous analysis in this region of the parameterspace.
6.2. Flexibly mounted cylinderTo assess the impact of the three-dimensional transition in the flexibly mountedcylinder case, two points have been selected in the parameter space, in the range ofrotation rates where the transition occurs for a rigidly mounted cylinder.
2 4 6 8
–0.10
–0.08
–0.06
–0.04
–0.02
0
0.02
0.04
Cx
(a () b)
Cx Cy
Cy
0 2 4 6 8 10
2 4 6 8 010z
–0.25
–0.20
–0.15
–0.10
–0.05
0
0.05
010z
–17.03
–17.02
–17.01
–17.00
–16.99
–16.98
–16.97
–16.96
–16.95
15.106
15.104
15.102
15.100
15.098
15.096
15.094
0 2 4 6 8 10
(d)(c)
FIGURE 25. (Colour online) Spanwise evolution of the (a,c) in-line and (b,d) cross-flowforce coefficients in the rigidly mounted cylinder case, for (a,b) α= 3.75 and (c,d) α= 4.0.The span-averaged value of the force coefficient is indicated by a dashed line, while thevalue issued from two-dimensional simulation is represented by a dash-dotted line.
The first point, (α,U?)= (3.75, 13), is located within the vibration region identifiedin § 3, in the area of maximum amplitudes of oscillation (figure 7a). A visualizationof the flow through instantaneous isosurfaces of spanwise vorticity issued from thethree-dimensional simulation is presented in figure 26. Contrary to the rigidly mountedcylinder case at the same rotation rate (figure 24c), it appears that the flow is strictlytwo-dimensional. The behaviour of the fluid–structure system, including the vibrationamplitude and frequency, the wake pattern and the fluid forces, is identical to thatissued from the two-dimensional simulation. Previous studies concerning non-rotatingcylinders have shown that the onset of three-dimensionality may be delayed when thecylinder oscillates transversely (e.g. Berger 1967; Leontini, Thompson & Hourigan2007); a similar phenomenon is observed in the present case of a rotating cylinder.Owing to the large amplitudes of the cylinder oscillations in the upper part of thevibration region, a comparable delay of the three-dimensional transition is likely tooccur across the entire vibration window.
The second point, (α,U?)= (4, 14), is chosen outside the vibration region definedon the basis of two-dimensional simulations. In this case, as also noted from the two-dimensional results in § 3, no vibration of the cylinder develops and the flow remainssteady. The flow pattern is identical to that observed in the rigidly mounted cylindercase (figure 24e,f ). Compared to the two-dimensional case, the equilibrium position of
x
y
z
FIGURE 26. (Colour online) Instantaneous isosurfaces of spanwise vorticity (ωz = ±0.2)for (α,U?)= (3.75, 13). Part of the computational domain is shown.
the cylinder is slightly displaced upwards due to the modification of the span-averagedvalue of Cy, equal to −16.98 in the three-dimensional case versus −17.01 under thetwo-dimensional flow assumption.
Therefore, the three-dimensional simulation results reported in this sectioncorroborate the previous two-dimensional analysis.
7. ConclusionsThe transverse flow-induced vibrations of a circular cylinder subjected to forced
rotation have been investigated by means of numerical simulations, at a Reynoldsnumber equal to 100. The impact of the symmetry breaking due to the imposedrotation on the phenomenon of vortex-induced vibrations, previously studied in theabsence of rotation, has been analysed over a wide range of reduced velocities androtation rates. The analysis was essentially based on two-dimensional simulations.Three-dimensional simulations were performed in selected points of the parameterspace in order to confirm the two-dimensional results and examine the potentialinfluence of the flow three-dimensional transition, which occurs at high rotation rates,for α ∈ [3.5, 3.75], when the cylinder is rigidly mounted. The principal findings ofthis work can be summarized as follows.
Large-amplitude vibrations through wake–body synchronization. The cylinder isfound to exhibit free oscillations from α = 0 up to a rotation rate close to 4.As a result, vibrations occur for α < 1.8 but also for α > 1.8, i.e. in a range ofrotation rates where the rotation cancels the vortex shedding in the rigidly mountedcylinder case. Under forced rotation, the vibrations may reach amplitudes close to1.9 cylinder diameters, thus three times larger than the maximum vibration amplitudeobserved for a non-rotating cylinder. For all rotation rates, the bell-shaped evolutionof the vibration amplitude as a function of the reduced velocity is comparable tothat noted for non-rotating cylinder VIV, although the vibration window may widensubstantially. Over the entire vibration region identified in the (α,U?) parameter space,the structural response and the wake unsteadiness appear to be synchronized. Thefree vibrations of the rotating cylinder thus occur under a condition of wake–bodysynchronization similar to the lock-in condition driving non-rotating cylinder VIV.
This behaviour contrasts with the galloping-like responses, which do not involve thelock-in condition and exhibit amplitudes that increase unboundedly with U?.
Novel wake patterns under forced rotation. Several flow patterns including novelwake topologies have been observed across the parameter space. In the absence ofvibration, the 2S pattern dominates for α < 1.8, while steady patterns characterizedby elongated layers of spanwise vorticity occur at higher rotation rates. Two steadypatterns have been identified: the D+ pattern (for α < 3.7 approximately), where thenegative-vorticity layer is located above the positive one, which results in a positivein-line force; and the D− pattern (for α > 3.7 approximately), which is characterizedby an inversion of the vorticity layers and Cx < 0. Owing to the three-dimensionaltransition, the vorticity layers of the D− pattern exhibit a spanwise undulation ofwavelength approximately equal to 1.7 cylinder diameters. Within the vibrationregion, the 2S pattern and its variations (C(2S) and P) dominate the low-rotation-ratearea, which corresponds to structural responses of moderate amplitudes and high tomoderate frequencies. Three wake patterns occur in the high-rotation-rate zone ofthe vibration region. The P + S pattern and a novel asymmetric pattern composedof a triplet of vortices and a single vortex shed per cycle, referred to as T + S,are associated with large-amplitude vibrations at moderate to low frequencies. TheT+ S pattern dominates the region of maximum-amplitude vibrations, which includesthe range of high rotation rates where the flow three-dimensional transition occursin the rigidly mounted cylinder case. Three-dimensional simulation results in thisregion indicate that the flow around the vibrating cylinder remains two-dimensional.In the low-frequency range, the flow exhibits a transverse undulation of thespanwise vorticity layers, without vortex shedding. This new wake topology isreferred to as the U pattern. Therefore, it is found that free oscillations of therotating cylinder may occur in the absence of vortex detachment. Consequently, theterminology flow-induced vibrations under wake–body synchronization is preferred to‘vortex-induced vibrations’ in order to designate the present structural responses.
Impact of rotation on force frequency content and phasing mechanism. Largemodulations of the fluid forces, including a substantial amplification of Cx, are notedwhen the cylinder vibrates. The type of wake pattern is found to relate to the deviationof Cy from the value observed in the rigidly mounted case due to the Magnus effect:the 2S, C(2S) and P patterns are associated with negative deviations, while positivedeviations are observed under the P + S, T + S and U patterns. As also noted inprevious works, in the absence of rotation, the transverse oscillation of the cylinderresults in a large third harmonic component in the cross-flow force, while thein-line force is dominated by a single frequency equal to twice the cross-flowfundamental frequency. The symmetry breaking caused by the rotation directlyimpacts the frequency content of the fluid forces: the ratio between the in-line andcross-flow force fundamental frequencies switches from 2 to 1, the in-line forcespectrum is dominated by the first harmonic component, while both second and thirdharmonic contributions arise in the cross-flow force spectrum. The imposed rotationalso influences the phasing between the cross-flow force and the structural response:the 180 phase jump occurring when the vibration frequency crosses the oscillatornatural frequency tends to drift towards higher reduced velocities as α increases anddisappears at high rotation rates. This phenomenon is associated with an alterationof the vibration frequency, which is found to globally decrease when the rotationrate increases, in relation with the strong variability of the effective added masscoefficient.
AcknowledgementThis work was performed using HPC resources from CALMIP (grant 2013-P1248).
REFERENCES
BADR, H. M., COUTANCEAU, M., DENNIS, S. C. R. & MÉNARD, C. 1990 Unsteady flow past arotating circular cylinder at Reynolds numbers 103 and 104. J. Fluid Mech. 220, 459–484.
BEARMAN, P. W. 1984 Vortex shedding from oscillating bluff bodies. Annu. Rev. Fluid Mech. 16,195–222.
BEARMAN, P. W. 2011 Circular cylinder wakes and vortex-induced vibrations. J. Fluids Struct. 27,648–658.
BEARMAN, P. W., GARTSHORE, I. S., MAULL, D. J. & PARKINSON, G. V. 1987 Experiments onflow-induced vibration of a square-section cylinder. J. Fluids Struct. 1, 19–34.
BERGER, E. 1967 Suppression of vortex shedding and turbulence behind oscillating cylinders. Phys.Fluids 10, S191.
BISHOP, R. E. D. & HASSAN, A. Y. 1964 The lift and drag forces on a circular cylinder oscillatingin a flowing fluid. Proc. R. Soc. A 277, 51–75.
BLACKBURN, H. M. & HENDERSON, R. D. 1999 A study of two-dimensional flow past an oscillatingcylinder. J. Fluid Mech. 385, 255–286.
BLEVINS, R. D. 1990 Flow-Induced Vibration. Van Nostrand Reinhold.BOURGUET, R., KARNIADAKIS, G. E. & TRIANTAFYLLOU, M. S. 2011a Lock-in of the vortex-
induced vibrations of a long tensioned beam in shear flow. J. Fluids Struct. 27, 838–847.BOURGUET, R., KARNIADAKIS, G. E. & TRIANTAFYLLOU, M. S. 2011b Vortex-induced vibrations
of a long flexible cylinder in shear flow. J. Fluid Mech. 677, 342–382.BRAZA, M., CHASSAING, P. & HA MINH, H. 1986 Numerical study and physical analysis of the
pressure and velocity fields in the near wake of a circular cylinder. J. Fluid Mech. 165,79–130.
BRIKA, D. & LANEVILLE, A. 1993 Vortex-induced vibrations of a long flexible circular cylinder.J. Fluid Mech. 250, 481–508.
CARBERRY, J., SHERIDAN, J. & ROCKWELL, D. 2001 Forces and wake modes of an oscillatingcylinder. J. Fluids Struct. 15, 523–532.
CARBERRY, J., SHERIDAN, J. & ROCKWELL, D. 2005 Controlled oscillations of a cylinder: forcesand wake modes. J. Fluid Mech. 538, 31–69.
CHEW, Y. T., CHENG, M. & LUO, S. C. 1995 A numerical study of flow past a rotating circularcylinder using a hybrid vortex scheme. J. Fluid Mech. 299, 35–71.
CORLESS, R. M. & PARKINSON, G. V. 1988 A model of the combined effects of vortex-inducedoscillation and galloping. J. Fluids Struct. 2, 203–220.
COUTANCEAU, M. & MÉNARD, C. 1985 Influence of rotation on the near-wake development behindan impulsively started circular cylinder. J. Fluid Mech. 158, 399–446.
DAHL, J. M., HOVER, F. S., TRIANTAFYLLOU, M. S., DONG, S. & KARNIADAKIS, G. E. 2007Resonant vibrations of bluff bodies cause multivortex shedding and high frequency forces.Phys. Rev. Lett. 99, 144503.
DAHL, J. M., HOVER, F. S., TRIANTAFYLLOU, M. S. & OAKLEY, O. H. 2010 Dual resonancein vortex-induced vibrations at subcritical and supercritical Reynolds numbers. J. Fluid Mech.643, 395–424.
EL AKOURY, R., BRAZA, M., PERRIN, R., HARRAN, G. & HOARAU, Y. 2008 The three-dimensional transition in the flow around a rotating cylinder. J. Fluid Mech. 607, 1–11.
EVANGELINOS, C. & KARNIADAKIS, G. E. 1999 Dynamics and flow structures in the turbulentwake of rigid and flexible cylinders subject to vortex-induced vibrations. J. Fluid Mech. 400,91–124.
GOVARDHAN, R. & WILLIAMSON, C. H. K. 2000 Modes of vortex formation and frequency responseof a freely vibrating cylinder. J. Fluid Mech. 420, 85–130.
HOVER, F. S., TECHET, A. H. & TRIANTAFYLLOU, M. S. 1998 Forces on oscillating uniform andtapered cylinders in crossflow. J. Fluid Mech. 363, 97–114.
JAUVTIS, N. & WILLIAMSON, C. H. K. 2004 The effect of two degrees of freedom on vortex-inducedvibration at low mass and damping. J. Fluid Mech. 509, 23–62.
JEON, D. & GHARIB, M. 2001 On circular cylinders undergoing two-degree-of-freedom forced motions.J. Fluids Struct. 15, 533–541.
KANG, S., CHOI, H. & LEE, S. 1999 Laminar flow past a rotating circular cylinder. Phys. Fluids11, 3312–3321.
KARNIADAKIS, G. E. & SHERWIN, S. 1999 Spectral/hp Element Methods for CFD. 1st edn. OxfordUniversity Press.
KHALAK, A. & WILLIAMSON, C. H. K. 1999 Motions, forces and mode transitions in vortex-inducedvibrations at low mass-damping. J. Fluids Struct. 13, 813–851.
KIM, J., KIM, D. & CHOI, H. 2001 An immersed-boundary finite-volume method for simulations offlow in complex geometries. J. Comput. Phys. 171, 132–150.
KLAMO, J. T., LEONARD, A. & ROSHKO, A. 2006 The effects of damping on the amplitude andfrequency response of a freely vibrating cylinder in cross-flow. J. Fluids Struct. 22, 845–856.
LEONTINI, J. S., STEWART, B. E., THOMPSON, M. C. & HOURIGAN, K. 2006 Wake state andenergy transitions of an oscillating cylinder at low Reynolds number. Phys. Fluids 18, 067101.
LEONTINI, J. S., THOMPSON, M. C. & HOURIGAN, K. 2007 Three-dimensional transition in thewake of a transversely oscillating cylinder. J. Fluid Mech. 577, 79–104.
LUCOR, D. & TRIANTAFYLLOU, M. S. 2008 Parametric study of a two degree-of-freedom cylindersubject to vortex-induced vibrations. J. Fluids Struct. 24, 1284–1293.
MEENA, J., SIDARTH, G. S., KHAN, M. H. & MITTAL, S. 2011 Three-dimensional instabilities inflow past a spinning and translating cylinder. In IUTAM Symposium on Bluff Body Flows,IIT-Kanpur, India (ed. S. Mittal & G. Biswas), pp. 75–78.
MITTAL, S. 2004 A finite element study of incompressible flows past oscillating cylinders andaerofoils. J. Appl. Mech. 71, 89–95.
MITTAL, S. & KUMAR, B. 2003 Flow past a rotating cylinder. J. Fluid Mech. 476, 303–334.MITTAL, S. & TEZDUYAR, T. E. 1992 A finite element study of incompressible flows past oscillating
cylinders and aerofoils. Intl J. Numer. Meth. Fluids 15, 1073–1118.MODARRES-SADEGHI, Y., MUKUNDAN, H., DAHL, J. M., HOVER, F. S. & TRIANTAFYLLOU, M. S.
2010 The effect of higher harmonic forces on fatigue life of marine risers. J. Sound Vib. 329,43–55.
MODI, V. J. 1997 Moving surface boundary-layer control: a review. J. Fluids Struct. 11, 627–663.NAUDASCHER, E. & ROCKWELL, D. 1994 Flow-Induced Vibrations: an Engineering Guide. Dover.NAVROSE, & MITTAL, S. 2013 Free vibrations of a cylinder: 3-D computations at Re = 1000.
J. Fluids Struct. 41, 109–118.NEMES, A., ZHAO, J., LO JACONO, D. & SHERIDAN, J. 2012 The interaction between flow-induced
vibration mechanisms of a square cylinder with varying angles of attack. J. Fluid Mech. 710,102–130.
NEWMAN, D. J. & KARNIADAKIS, G. E. 1997 A direct numerical simulation study of flow past afreely vibrating cable. J. Fluid Mech. 344, 95–136.
PAÏDOUSSIS, M. P., PRICE, S. J. & DE LANGRE, E. 2010 Fluid–Structure Interactions: Cross-Flow-Induced Instabilities. Cambridge University Press.
PRALITS, J. O., BRANDT, L. & GIANNETTI, F. 2010 Instability and sensitivity of the flow arounda rotating circular cylinder. J. Fluid Mech. 650, 513–536.
PRALITS, J. O., GIANNETTI, F. & BRANDT, L. 2013 Three-dimensional instability of the flow arounda rotating circular cylinder. J. Fluid Mech. 730, 5–18.
PRANDTL, L. 1926 Application of the ‘Magnus effect’ to the wind propulsion of ships. NACA Tech.Mem. 367.
PRASANTH, T. K. & MITTAL, S. 2008 Vortex-induced vibrations of a circular cylinder at lowReynolds numbers. J. Fluid Mech. 594, 463–491.
RAGHAVAN, K. & BERNITSAS, M. M. 2011 Experimental investigation of Reynolds number effecton vortex-induced vibration of rigid circular cylinder on elastic supports. Ocean Engng 38,719–731.
RAO, A., LEONTINI, J. S., THOMPSON, M. C. & HOURIGAN, K. 2013a Three-dimensionality inthe wake of a rapidly rotating cylinder in uniform flow. J. Fluid Mech. 730, 379–391.
RAO, A., LEONTINI, J. S., THOMPSON, M. C. & HOURIGAN, K. 2013b Three-dimensionality inthe wake of a rotating cylinder in a uniform flow. J. Fluid Mech. 717, 1–29.
SARPKAYA, T. 1979 Vortex-induced oscillations: a selective review. Trans. ASME: J. Appl. Mech.46, 241–258.
SARPKAYA, T. 2004 A critical review of the intrinsic nature of vortex-induced vibrations. J. FluidsStruct. 19, 389–447.
SHEN, L., CHAN, E.-S. & LIN, P. 2009 Calculation of hydrodynamic forces acting on a submergedmoving object using immersed boundary method. Comput. Fluids 38, 691–702.
SHIELS, D., LEONARD, A. & ROSHKO, A. 2001 Flow-induced vibration of a circular cylinder atlimiting structural parameters. J. Fluids Struct. 15, 3–21.
SINGH, S. P. & MITTAL, S. 2005 Vortex-induced oscillations at low Reynolds numbers: hysteresisand vortex-shedding modes. J. Fluids Struct. 20, 1085–1104.
STANSBY, P. K. & RAINEY, R. C. T. 2001 On the orbital response of a rotating cylinder in acurrent. J. Fluid Mech. 439, 87–108.
STANSBY, P. K. & SLAOUTI, A. 1993 Simulation of vortex shedding including blockage by therandom-vortex and other methods. Intl J. Numer. Meth. Fluids 17, 1003–1013.
STOJKOVIC, D., BREUER, M. & DURST, F. 2002 Effect of high rotation rates on the laminar flowaround a circular cylinder. Phys. Fluids 14, 3160–3178.
STOJKOVIC, D., SCHÖN, P., BREUER, M. & DURST, F. 2003 On the new vortex shedding modepast a rotating circular cylinder. Phys. Fluids 15, 1257–1260.
VANDIVER, J. K., JAISWAL, V. & JHINGRAN, V. 2009 Insights on vortex-induced, travelling waveson long risers. J. Fluids Struct. 25, 641–653.
VIKESTAD, K., VANDIVER, J. K. & LARSEN, C. M. 2000 Added mass and oscillation frequencyfor a circular cylinder subjected to vortex-induced vibrations and external disturbance.J. Fluids Struct. 14, 1071–1088.
WANG, X. Q., SO, R. M. C. & CHAN, K. T. 2003 A nonlinear fluid force model for vortex-inducedvibration of an elastic cylinder. J. Sound Vib. 260, 287–305.
WILLIAMSON, C. H. K. 1988 The existence of two stages in the transition to three-dimensionalityof a cylinder wake. Phys. Fluids 31, 3165–3168.
WILLIAMSON, C. H. K. & GOVARDHAN, R. 2004 Vortex-induced vibrations. Annu. Rev. Fluid Mech.36, 413–455.
WILLIAMSON, C. H. K. & ROSHKO, A. 1988 Vortex formation in the wake of an oscillating cylinder.J. Fluids Struct. 2, 355–381.
WU, X., GE, F. & HONG, Y. 2012 A review of recent studies on vortex-induced vibrations of longslender cylinders. J. Fluids Struct. 28, 292–308.
YOGESWARAN, V. & MITTAL, S. 2011 Vortex-induced and galloping response of a rotating circularcylinder. In IUTAM Symposium on Bluff Body Flows, IIT-Kanpur, India (ed. S. Mittal &G. Biswas), pp. 153–156.
ZHAO, M., CHENG, L. & ZHOU, T. 2013 Numerical simulation of vortex-induced vibration of asquare cylinder at a low Reynolds number. Phys. Fluids 25, 023603.
ZHOU, C. Y., SO, R. M. C. & LAM, K. 1999 Vortex-induced vibrations of an elastic circular cylinder.J. Fluids Struct. 13, 165–189.